Network Simplex Algorithm Research Articles

Applying a Newton Method to Strictly Convex Separable Network Quadratic Programs

By introducing quadratic penalty terms, a strictly convex separable network quadratic program can be reduced to an unconstrained optimization problem whose objective is a continuously differentiable piecewise quadratic function. A recently developed nonsmooth version of Newton's method is applied to the reduced problem. The generalized Newton direction is computed by an iterative procedure which exploits the special network data structures that originated from the network simplex method. New features of the algorithm include the use of min-max bases and a dynamic strategy in computation of the Newton directions. Some preliminary computational results are presented. The results suggest the use of warm start instead of cold start.

Goal network programs: A specialized algorithm and an application

This paper presents a specialized network procedure for the solution of pure goal network programs with preemptive priorities. The specialization solves such goal network programs efficiently since it requires a modification only in the pricing rule of the network simplex algorithm. The algorithm is used to solve multiple goal network programs that arise in the assignment of naval personnel to jobs. Computational experience indicates that the specialization dominates the sequential linear goal programming procedure.

Strongly polynomial dual simplex methods for the maximum flow problem

This paper presents dual network simplex algorithms that require at most 2nm pivots and O(n2m) time for solving a maximum flow problem on a network ofn nodes andm arcs. Refined implementations of these algorithms and a related simplex variant that is not strictly speaking a dual simplex algorithm are shown to have a complexity of O(n3). The algorithms are based on the concept of apreflow and depend upon the use of node labels that are underestimates of the distances from the nodes to the sink node in the extended residual graph associated with the current flow. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Multifleet routing and multistop flight scheduling for schedule perturbation

Efficient and effective incidental scheduling techniques for schedule perturbation are essential to an airline carrier's operations. This research aims at developing a framework to assist carriers in fleet routing and flight scheduling for schedule perturbations in the operations of multifleet and multistop flights. The framework is based on a basic multifleet schedule perturbation model constructed as a timespace network from which strategic models are developed to research incidental scheduling. These network models are formulated as multiple commodity network flow problems. Lagrangian relaxation with subgradient methods accompanied by the network simplex method, a Lagrangian heuristic and a modified subgradient method are developed to solve the problems. A case study regarding the international operations of a major Taiwan airline carrier is presented.

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call the minimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks with n nodes, m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n log n) ) time, where Δ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n 2 ) pivots and O(nm + n 2 log n) time.

A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2m lognC, n 2m2 logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm 2 logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).

Dynamic trees as search trees via euler tours, applied to the network simplex algorithm

Thedynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of dynamic trees, values must be combined along paths. For other applications, values must be combined over entire trees. For the latter situation, an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(logn) per tree operation, wheren is the number of tree vertices. We apply this representation to the implementation of two versions of the network simplex algorithm, resulting in a time of O(logn) per pivot, wheren is the number of vertices in the problem network.

A new strongly polynomial dual network simplex algorithm

This paper presents a new dual network simplex algorithm for the minimum cost network flow problem. The algorithm works directly on the original capacitated network and runs in O(mn(m +n logn) logn) time for the network withn nodes andm arcs. This complexity is better than the complexity of Orlin, Plotkin and Tardos’ (1993) dual network simplex algorithm by a factor ofm/n.

On strongly polynomial dual simplex algorithms for the maximum flow problem

Several pivot rules for the dual network simplex algorithm that enable it to solve a maximum flow problem on an n-node, m-arc network in at most 2nm pivots and O(n 2 m) time are presented. These rules are based on the concept of a preflow and depend upon the use of node labels which are either the lengths of a shortest pseudoaugmenting path from those nodes to the sink node or valid underestimates of those lengths. Extended versions of our algorithms are shown to solve an important class of parametric maximum flow problems with no increase in the worst-case pivot and time bounds of these algorithms.

Flows on hypergraphs

We consider the capacitated minimum cost flow problem on directed hypergraphs. We define spanning hypertrees so generalizing the spanning tree of a standard graph, and show that, like in the standard and in the generalized minimum cost flow problems, a correspondence exists between bases and spanning hypertrees. Then, we show that, like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems, most of the computations performed at each pivot operation have direct hypergraph interpretations.

An integrated model for fleet routing, flight scheduling and aircraft rental planning

Fleet routing, flight scheduling and aircraft renting playa key role in airline carrier operations. This research rums at developmg a model to assist carriers in drafting flight timetables, providing better fleet routes, and making rental plans when market demand conditions are expected to change in the near future. The model is constructed as a time-space network and formulated as a network flow problem with side constraints. A Lagrangian-based algonthm, based on Lagrangian relaxation, a subgradient method the network simplex method and a Lagrangian heuristic, is developed to solve the problem. To show how to apply the model in the real world, a case study regarding the operations of a major Taiwan airline carrier is demonstrated.

Solving Multicommodity Flow Problems with a Primal Embedded Network Simplex Algorithm

This article describes the authors’ experience solving large multicommodity flow problems with an embedded network simplex algorithm augmented with a fast-start heuristic for choosing an initial basis. The heuristic makes successive capacity allocations in an attempt to find a feasible initial basis. Our implementation of the heuristic makes use of piece-wise linear convex costs. The efficacy of our heuristic, and of the embedded network simplex method, is demonstrated on large publicly available multicommodity flow problems. Comparisons with other published computational results are given.

Airline Scheduling for the Temporary Closure of Airports

The poor scheduling of flights or a fleet resulting from temporary closure of airports may cause a substantial loss of profit and decreased levels of service for airline carriers. This research develops a framework in order to help carriers handle schedule perturbations resulting from the temporary closure of airports. The framework is based on a basic model constructed as a time-space network from which several strategic network models are developed for scheduling. These network models are formulated as pure network flow problems or network flow problems with side constraints. The former are solved using the network simplex method whereas the latter are solved using a Lagrangian relaxation-based algorithm. To test how well the model may be applied in the real world, a case study regarding the international operations of a major Taiwan airline was performed. The results show that the framework could be useful in actual operations.

A decision support framework for handling schedule perturbation

Perturbations of flight schedules may occur everyday. Poor scheduling of flights may result in a substantial loss of profit and decreased levels of service for air carriers. This research aims at developing a framework to help carriers in handling schedule perturbations caused by the breakdown of aircraft. The framework is based on a basic schedule perturbation model constructed as a dynamic network from which several perturbed network models are developed for scheduling following incidents. These network models are formulated as pure network flow problems or network flow problems with side constraints. The former are solved using the network simplex method while the latter are solved using Lagrangian relaxation with subgradient methods. To show how to apply the framework in the real world, a case study focusing on the operations of a major Taiwan air carrier in East Asia is presented.

A decision support framework for multi-fleet routing and multi-stop flight scheduling

Fleet routing and flight scheduling are important in airline carrier operations. Ineffective and inefficient fleet routing and flight scheduling will result in a substantial loss of profits. This research aims at developing a framework to help carriers adjust their draft timetables and fleet routes, when market demand conditions are expected to change in the near future. The framework is based on a basic model, constructed as a multi-fleet time-space network from which several strategic models are developed, to help carriers in fleet routing and flight scheduling. These models are formulated as multiple commodity network flow problems. The Lagrangian relaxation accompanied by the network simplex method, a Lagrangian heuristic and a modified sub-gradient method are suggested to solve the problems. A flow decomposition algorithm is also suggested in order to trace every aircraft route. To show how to apply this framework in the real world, a case study regarding the international operations of a major Taiwan airline was performed. The results show that, the framework would be useful for actual operations.

Performance driven spacing algorithms using attractive and repulsive constraints for submicron LSI's

This paper proposes one-dimensional spacing algorithms that minimize [maximize] the degree of separation [proximity] among the specified elements of a given integrated circuit layout. Number of problems on chip and MCM systems such as poor performance, higher crosstalk, lower yield etc. are related to the degree of separation [proximity]. The proposed algorithms utilize the attractive [repulsive] constraints to shrink [expand] the distances among the specified elements while keeping the layout free of any design rule errors. The spacing problem is reduced to a parametric linear programming problem and a network simplex algorithm is proposed for solving it. The proposed algorithms are implemented into a system called PERFECT for performance enhancement and crosstalk reduction. Given a routed design, PERFECT minimizes delay due to the coupled capacitance, which could contribute as much as 50-75% to the interconnect delay in near future. PERFECT utilizes the repulsive constraints between interconnect segments for the purpose. The experimental results show a significant delay improvement and the crosstalk reduction for submicron technologies. >

A Method of Determining Target Configurations for Power System Restoration Using a Mixed Integer Programming Approach

AbstractWhen electricity supply is interrupted due to a fault, it is essential that the power system be restored promptly according to an adequately planned restoration procedure. A problem of obtaining an appropriate target system for restoration is referred to as a power system restoration problem. The authors have already proposed a very efficient method based on the network simplex method which is in the category of mathematical programming approaches. Although the method is effective for most system conditions, it is likely to give an undesirable result in the case of abnormal network conditions.This paper proposes an effective and computationally fast solution algorithm based on the mixed integer programming approach, which can resolve the shortcoming of the former method The solution speed is improved greatly by incorporating the, restoration strategies of system operators. The following assumptions are postulated in this approach: (1)0–1 variables are allocated to branches and loads to present their status; (2) continuous variables are taken to represent branch flows: and (3) costs are assigned to branches to represent their priorities during restorative operations and power flow limits. The proposed approach can be used to complement the former approach.The new approach has been applied to restoration problems of practical size, and simulation results demonstrate its advantage over the former method for intricate operating conditions.

A generator and a simplex solver for network piecewise linear programs

This is a brief report on our recent work in network piecewise linear programming (NPLP), and it consists of two parts. In the first park, we describe a generator for NPLP problems which is derived from the classical network linear program generator NETGEN. The generator creates networks of the same topological structures as NETGEN, but each arc is associated with a convex piecewise linear cost. The purpose of this program is to provide a set of standard test problems which can be used to compare the performance of various algorithms for NPLP. In the second part, we introduce a network simplex method that directly solves a network piecewise linear program without reformulating it as a network linear program of higher dimension. Forty benchmark NPLP problems are solved by this method and a reformulation method. The computational results are in favor of the direct method and show that solving an NPLP problem is not much harder than solving a network linear program of the same dimension.

A practical anti-degeneracy row selection technique in network linear programming

A characteristic feature of the primal network simplex algorithm (NSA) is that it usually makes a large number of degenerate iterations. Though cycling and even stalling can be avoided by recently introduced pivot rules for NSA, the practical efficiency of these rules is not known yet. For the case when the simplex algorithm is used to solve the continuous linear programming (LP) problem there exists a practical anti-cycling procedure that proved to be efficient. It is based on an expanding relaxation of the individual bound on the variables. In this paper we discuss the adaptation of this method to NSA, taking advantage of the special integer nature of network problems. We also give an account of our experience with these ideas as they are experimentally implemented in the MINET network LP solver. Reductions of CPU time have been achieved on a smaller set of specially structured real-life problems.

Studies of lexicography in the generalized network simplex method

This paper introduces an analytical approach for studying lexicography in generalized network problems. The equations obtained can help us to understand and to extend the existing theory. First, it is verified that all nonzero elements have the same sign in each row vector of a basis inverse for a generalized network (GN) problem with positive multipliers. However, this property does not necessarily hold when there exist negative multipliers. Second, we developed a strategy to select the dropping arc in the GN simplex algorithm when addressing GN problems with positive andnegative multipliers. This strategy is also based on lexicography and requires performing some comparisons. However, the values to be compared are already known since they can be obtained as a by-product of the calculations necessary to compute the basis representation of the entering arc. Consequently, the computational effort per pivot step isO(n) in the worst case. This worst case effort is the same as that required by the strongly convergent rules for selecting the dropping arc in the method of strong convergence.